520 
Mr. Gompertz on the nature of the function 
q •* 
=zd , and H =g, the equation will stand Lj, = d. g]^ , and 
-a m 
==7^7'’ ^ observe that when q is 
affirmative, and x ( e) negative, that x (g) is negative. The 
equation == ^ may he written in general x (L^) = x(d) ± 
the positive number whose common logarithm is {x® (g) + 
.r X (^ ) } , the upper or under sign to be taken according as the 
logarithm of g is positive or negative, X® standing for the cha- 
racteristic of a second logarithm ; that is, the logarithm of a 
logarithm, \{q) —1 x X (p). A" (g) = X» (s) — a.X (q) = 
. X(^) = X (m) — X ( 1 —p) — a X (</); also \(d) = 
^ (L ) - 
m 
Art. 7. Applying this to the interpolation of the North- 
ampton table, I observe that taking a= is and r= lo from 
that table, I find X (LJ-X (L^ ^ , 0566 = m, X (L^ J— 
^ (!'<.+ 2r)= .0745, X — X (L^^3,.) = , 0915, and 
^ (La + ,r) = ,1228 ; now if these numbers were 
in geometrical progression, whose ratio is p, we should have 
respectively m = ,0566 ; mp == ,0745 ; ?np^ =,0915 ; mp^= 
,1228. No value of p can be assumed which will make these 
equations accurately true ; but the numbers are such that p 
may be assumed, so that the equation shall be nearly true ; 
for resuming the first and last equations we have p^ = —^; 
.*. logarithm of p = ^(logarithm of 1228 — logarithm of 
566) = ,11213, X(^) = ,011213 and /)= 1,2944. And to 
examine how near this is to the thing required, continually 
to the logarithm of ,0566 namely 2,75282, adding ,11213 
which is the logarithm of p, we have respectively for the 
